Advanced Quantum Theory WS 2015/16
Problem Set 5 Due: 19/20 November 2015
Problem 14 Perturbed Harmonic Oscillator (Oral) A harmonic oscillator is subject to the perturbation
Vˆ(t) =
√3
2 ~ωΘ(t)(|2i h1|+|1i h2|), (1)
where Θ(t) is the Heaviside step function. Given that the oscillator is prepared in the state|n= 2i fort≤0, compute the probability amplitudes of the various unperturbed oscillator stateshn|ψ(t)i at timet >0,
(a) by solving the probability amplitude differential equations directly
(Hint: It is useful to first scale the differential equations by defining a dimensionless time τ =ωt/2.)
(b) by computing the eigenvectors and eigenvalues of the Hamiltonian ˆH0+ ˆV.
(c) Make a sketch of the lowest few energy eigenvalues of the system Hamiltonian fort < 0 and t >0, side by side.
Problem 15 Rotational Motion of a Rigid Body (Written)
Consider a diatomic molecule composed of two identical atoms rotating about its center of mass in a space-fixed coordinate system. For simplicity assume that the distance between the atoms is fixed.1 The Hamiltonian of the molecule is:
Hˆ = Lˆ2
2I, (2)
whereI is the moment of inertia of the molecule with respect to the rotational axis.
(a) What are the energy eigenvalues and eigenfunctions of the molecule and what are the dege- neracies of the energies? Determine the energy difference between two rotational levelsl and l+ 1.
(b) Assume the molecule is prepared in the state
ψ(θ, φ) =A(cos2θ+ sin2θcos 2φ), (3) whereA is a normalization constant.
(i) What are the probabilities of measuring the values 6~2, 2~2 and 0~2 forLˆ2?
(ii) What is the probability of measuring simultaneously the pair (6~2,−2~) forLˆ2 and ˆLz? Hint: Expressψ(θ, φ) in terms of the spherical harmonics Ylm(θ, φ) and determineA.
1In the literature such a system is also called arigid rotor.
1
Problem 16 Spin-12 in a Magnetic Field (Oral)
A particle with spin 1/2 and magnetic moment µis placed in a magnetic field
B=B0ˆz+B1cos(wt)ˆx−B1sin(wt)ˆy (4) which is often employed in magnetic resonance experiments.
Assume that the particle has spin up along +z -axis at t= 0 (ms= +1/2). Derive the probability to find the particle with spin down (ms=−1/2) at time t >0
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